3.88 \(\int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx\)

Optimal. Leaf size=262 \[ \frac{d^2 (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{f (m+1)}+\frac{2 d e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+2 n+1} \]

[Out]

(d^2*(f*x)^(1 + m)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1
+ m)/(2*n), -((c*x^(2*n))/a)])/(f*(1 + m)*(1 + (c*x^(2*n))/a)^p) + (2*d*e*x^(1
+ n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + n)/(2*n), -p, (1 + m +
3*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + n)*(1 + (c*x^(2*n))/a)^p) + (e^2*x^(1
+ 2*n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 +
m + 4*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + 2*n)*(1 + (c*x^(2*n))/a)^p)

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Rubi [A]  time = 0.348406, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d^2 (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{f (m+1)}+\frac{2 d e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+2 n+1} \]

Antiderivative was successfully verified.

[In]  Int[(f*x)^m*(d + e*x^n)^2*(a + c*x^(2*n))^p,x]

[Out]

(d^2*(f*x)^(1 + m)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1
+ m)/(2*n), -((c*x^(2*n))/a)])/(f*(1 + m)*(1 + (c*x^(2*n))/a)^p) + (2*d*e*x^(1
+ n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + n)/(2*n), -p, (1 + m +
3*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + n)*(1 + (c*x^(2*n))/a)^p) + (e^2*x^(1
+ 2*n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 +
m + 4*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + 2*n)*(1 + (c*x^(2*n))/a)^p)

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Rubi in Sympy [A]  time = 46.7903, size = 228, normalized size = 0.87 \[ \frac{d^{2} \left (f x\right )^{m + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{2 n} \\ 1 + \frac{m + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{f \left (m + 1\right )} + \frac{2 d e x^{n} \left (f x\right )^{- n} \left (f x\right )^{m + n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + n + 1}{2 n} \\ \frac{m + 3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{f \left (m + n + 1\right )} + \frac{e^{2} x^{2 n} \left (f x\right )^{- 2 n} \left (f x\right )^{m + 2 n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 2 n + 1}{2 n} \\ \frac{m + 4 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{f \left (m + 2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x)**m*(d+e*x**n)**2*(a+c*x**(2*n))**p,x)

[Out]

d**2*(f*x)**(m + 1)*(1 + c*x**(2*n)/a)**(-p)*(a + c*x**(2*n))**p*hyper((-p, (m +
1)/(2*n)), (1 + (m + 1)/(2*n),), -c*x**(2*n)/a)/(f*(m + 1)) + 2*d*e*x**n*(f*x)*
*(-n)*(f*x)**(m + n + 1)*(1 + c*x**(2*n)/a)**(-p)*(a + c*x**(2*n))**p*hyper((-p,
(m + n + 1)/(2*n)), ((m + 3*n + 1)/(2*n),), -c*x**(2*n)/a)/(f*(m + n + 1)) + e*
*2*x**(2*n)*(f*x)**(-2*n)*(f*x)**(m + 2*n + 1)*(1 + c*x**(2*n)/a)**(-p)*(a + c*x
**(2*n))**p*hyper((-p, (m + 2*n + 1)/(2*n)), ((m + 4*n + 1)/(2*n),), -c*x**(2*n)
/a)/(f*(m + 2*n + 1))

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Mathematica [A]  time = 0.335145, size = 189, normalized size = 0.72 \[ x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (\frac{d^2 \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{m+1}+e x^n \left (\frac{2 d \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1}+\frac{e x^n \, _2F_1\left (\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+2 n+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(f*x)^m*(d + e*x^n)^2*(a + c*x^(2*n))^p,x]

[Out]

(x*(f*x)^m*(a + c*x^(2*n))^p*((d^2*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 +
m)/(2*n), -((c*x^(2*n))/a)])/(1 + m) + e*x^n*((2*d*Hypergeometric2F1[(1 + m + n
)/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + n) + (e*x^n*Hyperg
eometric2F1[(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a)])/(1
+ m + 2*n))))/(1 + (c*x^(2*n))/a)^p

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \[ \int \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x)

[Out]

int((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{n} + d\right )}^{2}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p*(f*x)^m,x, algorithm="maxima")

[Out]

integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p*(f*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}\right )}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p*(f*x)^m,x, algorithm="fricas")

[Out]

integral((e^2*x^(2*n) + 2*d*e*x^n + d^2)*(c*x^(2*n) + a)^p*(f*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x)**m*(d+e*x**n)**2*(a+c*x**(2*n))**p,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p*(f*x)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError